Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle

Abstract

A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space $E$, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a program to generalize harmonic analysis in Euclidean spaces. This gives us a new, geometric way of viewing the Dunkl operators. In particular, we present a new proof of the commutativity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero.

References in corpus (5)

• Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators
• Geometry of Quantum Principal Bundles I
• Geometry of Quantum Principal Bundles II-Extended Version
• Characteristic Classes of Quantum Principal Bundles
• Relativity grupoid insted of relativity group

Cited by in corpus (1)

• Dunkl Operators for Arbitrary Finite Groups